Laguerre - Maple Help

Laguerre ODEs

Description

 • The general form of the Laguerre ODE is given by the following:
 > Laguerre_ode := x*diff(y(x),x,x)+(a+1-x)*diff(y(x),x)+lambda*y(x) = 0;
 ${\mathrm{Laguerre_ode}}{≔}{x}{}\left(\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){+}\left({a}{+}{1}{-}{x}\right){}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){+}{\mathrm{\lambda }}{}{y}{}\left({x}\right){=}{0}$ (1)
 See Iyanaga and Kawada, "Encyclopedic Dictionary of Mathematics", p. 1481. The solution to this type of ODE can be expressed in terms of the WhittakerW and WhittakerM functions.

Examples

 > $\mathrm{with}\left(\mathrm{DEtools},\mathrm{odeadvisor}\right)$
 $\left[{\mathrm{odeadvisor}}\right]$ (2)
 > $\mathrm{odeadvisor}\left(\mathrm{Laguerre_ode}\right)$
 $\left[{\mathrm{_Laguerre}}\right]$ (3)
 > $\mathrm{dsolve}\left(\mathrm{Laguerre_ode}\right)$
 ${y}{}\left({x}\right){=}{\mathrm{_C1}}{}{\mathrm{KummerM}}{}\left({-}{\mathrm{\lambda }}{,}{a}{+}{1}{,}{x}\right){+}{\mathrm{_C2}}{}{\mathrm{KummerU}}{}\left({-}{\mathrm{\lambda }}{,}{a}{+}{1}{,}{x}\right)$ (4)